Orbital stability

In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form $u(x,t)=e^{-i\omega t}\varphi (x)$ is said to be orbitally stable if any solution with the initial data sufficiently close to $\varphi (x)$ forever remains in a given small neighborhood of the trajectory of $e^{-i\omega t}\varphi (x).$

Formal definition

Formal definition is as follows. Consider the dynamical system

$i\frac{du}{dt}=A(u),u(t)\in X,t\in ℝ,$

with $X$ a Banach space over $ℂ$, and $A:X\to X$. We assume that the system is $\mathrm{U}(1)$-invariant, so that $A(e^{is}u)=e^{is}A(u)$ for any $u\in X$ and any $s\in ℝ$.

Assume that $\omega \varphi =A(\varphi )$, so that $u(t)=e^{-i\omega t}\varphi$ is a solution to the dynamical system. We call such solution a solitary wave.

We say that the solitary wave $e^{-i\omega t}\varphi$ is orbitally stable if for any $ϵ>0$ there is $\delta >0$ such that for any $v_0\in X$ with $\Vert \varphi -v_0{\Vert }_X<\delta$ there is a solution $v(t)$ defined for all $t\ge 0$ such that $v(0)=v_0$, and such that this solution satisfies

$\underset{t\ge 0}{\sup }\underset{s\in ℝ}{\inf }\Vert v(t)-e^{is}\varphi {\Vert }_X<ϵ.$

Example

According to , the solitary wave solution $e^{-i\omega t}{\varphi }_{\omega }(x)$ to the nonlinear Schrödinger equation

$i\frac{\partial }{\partial t}u=-\frac{{\partial }^2}{\partial x^2}u+g(|u{|}^2)u,u(x,t)\in ℂ,x\in ℝ,t\in ℝ,$

where $g$ is a smooth real-valued function, is orbitally stable if the Vakhitov–Kolokolov stability criterion is satisfied:

$\frac{d}{d\omega }Q({\varphi }_{\omega })<0,$

where

$Q(u)=\frac{1}{2}\underset{ℝ}{\int }|u(x,t){|}^{2}dx$

is the charge of the solution $u(x,t)$, which is conserved in time (at least if the solution $u(x,t)$ is sufficiently smooth).

It was also shown, that if $\frac{d}{d\omega }Q(\omega )<0$ at a particular value of $\omega$, then the solitary wave $e^{-i\omega t}{\varphi }_{\omega }(x)$ is Lyapunov stable, with the Lyapunov function given by $L(u)=E(u)-\omega Q(u)+\mathrm{\Gamma }(Q(u)-Q({\varphi }_{\omega }){)}^2$, where $E(u)=\frac{1}{2}\underset{ℝ}{\int }({\left|\frac{\partial u}{\partial x}\right|}^2+G(|u{|}^2))dx$ is the energy of a solution $u(x,t)$, with $G(y)={\int }_0^{y}g(z)dz$ the antiderivative of $g$, as long as the constant $\mathrm{\Gamma }>0$ is chosen sufficiently large.

See also

• Stability theory
  • Asymptotic stability
  • Linear stability
  • Lyapunov stability
  • Vakhitov−Kolokolov stability criterion